Projective Geometry

A geometry which begins with the ordinary points and lines of
Euclidean plane geometry, and adds an ideal line consisting of ideal
points which are considered the intersections of parallel lines. Thus each
equivalence class of parallel lines contains one of these ideal points, which is
defined in projective geometry as the intersection of these parallel lines.
The ordinary and ideal points, together, comprise the real projective plane.
Then, for three dimensions, we add an ideal plane consisting of one ideal line
for each equivalence class of parallel planes, each of which is the intersection
of those planes.

Axioms of Real Projective Plane Geometry

There exists at least one line.

On each line there exist at least three points.

Not all points lie on the same line.

Two distinct points lie on (and determine) exactly one line.

Two distinct lines meet in (and determine) exactly one point.

There is a one-to-one correspondence between the real numbers and all
but one point of a line.

The Principle of Plane Duality

Statement or phrase

Dual of the statement or phrase

point

line

a point lies on a line

a line lies on (i.e. contains) a point

There exists at least one line

There exists at least one point

On each line there are at least three points

Through each point there are at least three lines

Not all points lie on the same line

Not all lines pass through the same point

Two distinct points determine a unique line

Two distinct lines determine a unique point

Three (or any number of) non-collinear points

Three (or any number of) non-concurrent lines

Self Dual: A polygon is a coplanar multiset (i.e. set
whose elements are not necessarily distinct) of points (vertices) and
lines (sides, or edges) such that each vertex is at the intersection of
exactly two sides, and each side contains exactly two vertices.

Self Dual: A polygon is a coplanar multiset of lines (sides, or
edges) and points (vertices) such that each side is determined by exactly
two vertices, and each vertex is determined by exactly two sides.

A polygon is inscribed in a figure when all of its vertices lie on the
figure.

A polygon is circumscribed about the figure when all of its sides are
tangent to the figure.

The Principle of Space Duality

Statement or phrase

Dual of the statement or phrase

point

plane

a line lies on (i.e. contains) a point

a line lies in a plane

two distinct points determine a unique line

two distinct planes determine (meet in) a unique line

three distinct points not all on the same line determine a unique
plane

three distinct planes not all containing the same line determine (meet
in) a unique point

if two lines lie on a unique plane, they determine in a unique point

if two lines lie on (i.e. contain) a unique point, they
determine a unique plane

a line and a plane not containing the line meet in a unique point

a line and a point not on the line determine a unique plane

a triangle is three coplanar lines that are determined by three
non-collinear points, which lie in a plane

the triangle's space dual is three concurrent lines that are
determined by three non-collinear planes, which meet in a point

The platonic solids are the
tetrahedron, cube, octahedron, dodecahedron, and icosahedron, which have
4, 6, 8, 12, and 20 faces, respectively.

The platonic solids are the
tetrahedron (self-dual), octahedron, cube, icosahedron, and
dodecahedron, which have 4, 6, 8, 12, and 20 vertices, respectively.

To make the dual terminology of a point lying on a line vs. a
line passing through a point more nearly parallel, some authorities use
the expression "incident to" to mean either one. For example, three
collinear points are said to be incident to a particular line, and three
concurrent lines are said to be incident to a particular point.

Measurements

You win some; you lose some. With projective geometry we gain the
ability to use duals, including statements about duals, and complete proofs
about duals to get two proofs in one. We also gain the simplification of
geometry in which parallel lines and intersecting lines are not distinguished;
quite the contrary, all pairs of lines in a plane intersect in exactly one
point.

But we lose some, too. Since we added a point to every line -- the
so-called "ideal" point -- we don't retain order, hence we lose measurement.
The sides of a triangle are no longer line segments contained "between" a pair
of points, because the whole idea of "between" is lost; a triangle is the three
lines, not just segments of lines.

It has been pointed out that the origin of the term "geometry" comes from geo
(Earth) and meter (measurement), so "projective geometry" is something of an
oxymoron, since all measurement is lost.

Conic Sections

Having lost all sense of measurement, the conic sections are
indistinguishable from one another in projective geometry. That's actually
a good thing, because it makes every theorem proved purely using projective
geometry that applies to, say, a circle much more general. Without further
work, the same theorem applies to an ellipse, a hyperbola, a parabola, and even
a pair of intersecting lines, which include parallel lines, since they intersect
as well! For that reason,
Pascal's Theorem and its dual,
Brianchon's Theorem apply equally to any conic section, and are a
generalization of Pappus' theorem,
which only applies to a pair of intersecting lines.